Advanced Microeconomic Theory II
Johan Stennek
Spring 2016
14 March 2016
PS C - Dynamic Games with Applications
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Divide & Choose .............................................................................................. 2
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Centipede ....................................................................................................... 2
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Take it or leave it............................................................................................. 3
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Two-Period Bargaining .................................................................................... 3
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Strategic investment (not to be discussed in class) ........................................... 4
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1 Divide & Choose
Two children are to split a cake. The first kid gets to split the cake in two pieces
and the second kid gets to choose which piece he wants. Both children want as
much cake as possible for themselves. This is understood by both.
For simplicity, you may assume that the first child cuts the cake into a left and a
right piece and that the left piece must be 25%, 50% or 75% of the cake.
1.
2.
3.
4.
5.
Write down the extensive form game tree!
Does this game have perfect information?
Does this game have complete information?
What is the appropriate solution concept?
Find all such equilibria!
If you have the time, you may also do the following exercises (not to be discussed
in class):
6. Write down the normal form!
7. Find the Nash equilibria!
2 Centipede
Arne and Bertil sit at a table. Arne has two piles of money in front of him, one
with €10 and the other with €5. Arne has two choices. Either he takes the larger
pile of money (€10) and gives the smaller to Bertil (€5). Or he pushes both piles
to Bertil, in which case the money doubles. Bertil has then the same choice. This
goes on for (say) 4 times. If no one has taken the money, both piles double again
and Arne gets €160 and Bertil gets €80.
1. Write this up as an extensive form game.
2. Solve it
3. What would you have done if you where Arne?
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3 Take it or leave it
A buyer and a seller meet to negotiate the price of a unique good. The buyer’s
valuation of the good is vB and the seller’s valuation of the good if he keeps it is
vS > vB . These valuations are common knowledge. The two parties are in a hurry
so they will have no time for haggling.
1. Assume that the seller gets to offer a price and that the buyer only can
accept or reject it. Will there be trade? If so, what will the price be? 2. Assume that the buyer gets to offer a price and that the seller only can
accept or reject it. Will there be trade? If so, what will the price be? 4 Two-Period Bargaining
Consider the same situation as above and assume that they have time for two
periods of negotiations. The buyer’s discount factor is d B and the seller’s is d S.
1. Assume that the buyer gets to offer a price in period 1 and that the seller
can make a counter offer in period 2. Will there be trade? If so, what will
the price be? When will they come to an agreement? Compare with the
outcome of “take-it-or-leave-it-bargaining”.
2. Assume that the seller gets to offer a price in period 1 and that the buyer
can make a counter offer in period 2. What will the price be? Compare
with the outcome of “take-it-or-leave-it-bargaining”. If you have the time, try also to solve the game for three periods (not to be
discussed in class).
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5 Strategic investment (not to be discussed in class)
Consider a monopoly firm with linear demand p = a - b × q and constant
marginal cost c. Recall that the monopolist will produce quantity
qm ( c) = 2×1b × ( a- c) . It will thus make profit
p m ( c) = (a - b ×qm ( c) - c) × qm ( c) - I ( c) ,
where I ( c) is the firm’s cost of investments. In particular, before production
starts, the firm can invest in modern machinery and reduce marginal cost. A
lower marginal cost requires a higher investment, i.e. I ' ( c) < 0 . (Notice that we
think of the investment decision as a choice of marginal cost.)
1. Derive the monopolist’s first-order condition for the optimal investment.
Make explicit use of the Envelope Theorem.
2. Explain what the monopolist’s gain from investment is.
Consider now the same market, but assume that there are 2 firms competing a la
Cournot. Let qi be firm i’s quantity. The price is given by the inverse demand
function, which is linear and given by p = a - b × ( q1 + q2 ) . Firm i’s marginal cost is
constant and denoted ci . Notice that firm 1’s equilibrium profit can be written
as:
p1* ( c1,c2 ) = ×
( 1* ( c1,c2 ) + q2* ( c1,c2 )) - c1 ×××q1* ( c1,c2 ) - I ( c1 )
×a - b ×q
where I ( c1 ) is the firm’s cost of investments. In particular, before production
starts, firm 1 can invest in modern machinery and reduce marginal cost. A lower
marginal cost requires a higher investment, i.e. I ' ( c) < 0 .
3. Derive the duopolist’s first-order condition for the optimal investment.
Make explicit use of the Envelope Theorem.
4. Explain what the duopolist’s gain from investment is.
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